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On the Sums of Compound Negative Binomial and Gamma Random Variables

Part of: Distribution theory - Probability Distribution theory

Published online by Cambridge University Press:  14 July 2016

P. Vellaisamy  and
N. S. Upadhye
P. Vellaisamy*
Affiliation:
Indian Institute of Technology Bombay
N. S. Upadhye*
Affiliation:
Indian Institute of Technology Bombay
*
∗Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India.
∗Postal address: Department of Mathematics, Indian Institute of Technology Bombay, Powai, Mumbai, 400 076, India.
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Abstract

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We study the convolution of compound negative binomial distributions with arbitrary parameters. The exact expression and also a random parameter representation are obtained. These results generalize some recent results in the literature. An application of these results to insurance mathematics is discussed. The sums of certain dependent compound Poisson variables are also studied. Using the connection between negative binomial and gamma distributions, we obtain a simple random parameter representation for the convolution of independent and weighted gamma variables with arbitrary parameters. Applications to the reliability of m-out-of-n:G systems and to the shortest path problem in graph theory are also discussed.

Keywords

Compound negative binomial distributionconvolutionrandom parameter representationcompound Poisson distributionsums of gamma random variablesPoisson process

MSC classification

Secondary: 60E05: Distributions 62E15: Exact distribution theory
Type
Research Article
Information
Journal of Applied Probability , Volume 46 , Issue 1 , March 2009 , pp. 272 - 283
Copyright
Copyright © Applied Probability Trust 2009 

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